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Theorem bcthlem2 25359

Description: Lemma for bcth 25363. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)

**Hypotheses**
Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
bcthlem.6	⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem.7	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
bcthlem.8	⊢ (𝜑 → 𝐶 ∈ 𝑋)
bcthlem.9	⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
bcthlem.10	⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
bcthlem.11	⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))

**Assertion**
Ref	Expression
bcthlem2	⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Distinct variable groups: 𝑘,𝑛,𝑟,𝑥,𝑧 𝐶,𝑟,𝑥 𝑔,𝑘,𝑛,𝑟,𝑥,𝑧,𝐷 𝑔,𝐹,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝐽,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝑀,𝑘,𝑛,𝑟,𝑥,𝑧 𝜑,𝑘,𝑛,𝑟,𝑥,𝑧 𝑥,𝑅 𝑔,𝑋,𝑘,𝑛,𝑟,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑔) 𝐶(𝑧,𝑔,𝑘,𝑛) 𝑅(𝑧,𝑔,𝑘,𝑛,𝑟)

**Proof of Theorem bcthlem2**
Step	Hyp	Ref	Expression
1		bcthlem.11	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
2		fvoveq1 7454	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑛 + 1)))
3		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
4		fveq2 6906	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑔‘𝑘) = (𝑔‘𝑛))
5	3, 4	oveq12d 7449	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘𝐹(𝑔‘𝑘)) = (𝑛𝐹(𝑔‘𝑛)))
6	2, 5	eleq12d 2835	. . . . . 6 ⊢ (𝑘 = 𝑛 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))))
7	6	rspccva 3621	. . . . 5 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
8	1, 7	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
9		bcthlem.9	. . . . . 6 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
10	9	ffvelcdmda 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))
11		bcth.2	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
12		bcthlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
13		bcthlem.5	. . . . . . 7 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
14	11, 12, 13	bcthlem1 25358	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
15	14	expr 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘𝑛) ∈ (𝑋 × ℝ⁺) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))))
16	10, 15	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
17	8, 16	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))
18		cmetmet 25320	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
19	12, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
20		metxmet 24344	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
22	11	mopntop 24450	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Top)
24		xp1st 8046	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋)
25		xp2nd 8047	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ⁺)
26	25	rpxrd 13078	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)
27	24, 26	jca 511	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*))
28		blssm 24428	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
29	28	3expb 1121	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
30	21, 27, 29	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
31		df-ov 7434	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
32		1st2nd2 8053	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (𝑔‘(𝑛 + 1)) = ⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
33	32	fveq2d 6910	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩))
34	31, 33	eqtr4id 2796	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
35	34	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
36	11	mopnuni 24451	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
37	21, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
38	37	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → 𝑋 = ∪ 𝐽)
39	30, 35, 38	3sstr3d 4038	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽)
40		eqid 2737	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	sscls 23064	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
42	23, 39, 41	syl2an2r 685	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
43		difss2 4138	. . . . . . . 8 ⊢ (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
44		sstr2 3990	. . . . . . . 8 ⊢ (((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
45	42, 43, 44	syl2im 40	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
46	45	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))))
47	46	ex 412	. . . . 5 ⊢ (𝜑 → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))))
48	47	3impd 1349	. . . 4 ⊢ (𝜑 → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
49	48	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
50	17, 49	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
51	50	ralrimiva 3146	1 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ⊆ wss 3951 ⟨cop 4632 ∪ cuni 4907 class class class wbr 5143 {copab 5205 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1^st c1st 8012 2^nd c2nd 8013 1c1 11156 + caddc 11158 ℝ^*cxr 11294 < clt 11295 / cdiv 11920 ℕcn 12266 ℝ⁺crp 13034 ∞Metcxmet 21349 Metcmet 21350 ballcbl 21351 MetOpencmopn 21354 Topctop 22899 Clsdccld 23024 clsccl 23026 CMetccmet 25288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cld 23027 df-cls 23029 df-cmet 25291

This theorem is referenced by: bcthlem3 25360 bcthlem4 25361

W3C validator